Sure!
Step 1: Conceptual Understanding
The first thing to know here is that you're being asked to construct a graph of a function where the limit as x approaches -4 from the left (x->-4-) is negative infinity and the limit as x approaches -4 from the right (x->-4+) is four.
Step 2: Function Selection
The simple function that meets these conditions is y = 1/(x+4). This function has an infinite discontinuity at x = -4, creating an asymptote at that value. When x approaches -4 from the right (x->-4+), the denominator gets closer to zero from the positive side and y goes towards positive infinity (which is not what we need). So consider the negative of this function, y = -1/(x+4). Now, as x approaches -4 from the right (x->-4+), y goes towards zero which is exactly what we need. And as x approaches -4 from the left (x->-4-), y goes towards negative infinity.
Step 3: Drawing the Graph
Here are the steps to draw the graph of y = -1/(x+4):
1. Draw Cartesian axes, label the x-axis as x and the y-axis as f(x). Mark the point (-4,0) on x-axis.
2. To the right of -4, the graph of -1/(x+4) is a hyperbola opening downwards and approaching the x-axis (y = 0) but never quite reaching it hence creating a horizontal asymptote.
3. To the left of -4, the graph of -1/(x+4) starts from negative infinity at x=-4 and rises but never crosses the x-axis hence creating a vertical asymptote.
4. Signify the asymptote at x = -4 with a red dotted vertical line labeled 'x=-4'.
5. You can select random points on either side of x = - 4 in case you would like to show specific values on y.
There you have it! This graph should clearly illustrate the required limit behaviors of the function. When x approaches -4 from the left or right, the y values either decrease without bound or approach zero respectively.